Optimal. Leaf size=24 \[ b \text {Int}\left (\frac {\csc \left (c+d x^2\right )}{x^2},x\right )-\frac {a}{x} \]
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Rubi [A] time = 0.02, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {a+b \csc \left (c+d x^2\right )}{x^2} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {a+b \csc \left (c+d x^2\right )}{x^2} \, dx &=\int \left (\frac {a}{x^2}+\frac {b \csc \left (c+d x^2\right )}{x^2}\right ) \, dx\\ &=-\frac {a}{x}+b \int \frac {\csc \left (c+d x^2\right )}{x^2} \, dx\\ \end {align*}
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Mathematica [A] time = 0.25, size = 0, normalized size = 0.00 \[ \int \frac {a+b \csc \left (c+d x^2\right )}{x^2} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.50, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b \csc \left (d x^{2} + c\right ) + a}{x^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {b \csc \left (d x^{2} + c\right ) + a}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 0, normalized size = 0.00 \[ \int \frac {a +b \csc \left (d \,x^{2}+c \right )}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.00, size = 0, normalized size = 0.00 \[ b {\left (\int \frac {\sin \left (d x^{2} + c\right )}{x^{2} \cos \left (d x^{2} + c\right )^{2} + x^{2} \sin \left (d x^{2} + c\right )^{2} + 2 \, x^{2} \cos \left (d x^{2} + c\right ) + x^{2}}\,{d x} + \int \frac {\sin \left (d x^{2} + c\right )}{x^{2} \cos \left (d x^{2} + c\right )^{2} + x^{2} \sin \left (d x^{2} + c\right )^{2} - 2 \, x^{2} \cos \left (d x^{2} + c\right ) + x^{2}}\,{d x}\right )} - \frac {a}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [A] time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {a+\frac {b}{\sin \left (d\,x^2+c\right )}}{x^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {a + b \csc {\left (c + d x^{2} \right )}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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